Problem: Simplify; express your answer in exponential form. Assume $y\neq 0, t\neq 0$. $\dfrac{{(y^{3}t^{4})^{3}}}{{(y^{-3}t^{-4})^{-4}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(y^{3}t^{4})^{3} = (y^{3})^{3}(t^{4})^{3}}$ On the left, we have ${y^{3}}$ to the exponent ${3}$ . Now ${3 \times 3 = 9}$ , so ${(y^{3})^{3} = y^{9}}$ Apply the ideas above to simplify the equation. $\dfrac{{(y^{3}t^{4})^{3}}}{{(y^{-3}t^{-4})^{-4}}} = \dfrac{{y^{9}t^{12}}}{{y^{12}t^{16}}}$ Break up the equation by variable and simplify. $\dfrac{{y^{9}t^{12}}}{{y^{12}t^{16}}} = \dfrac{{y^{9}}}{{y^{12}}} \cdot \dfrac{{t^{12}}}{{t^{16}}} = y^{{9} - {12}} \cdot t^{{12} - {16}} = y^{-3}t^{-4}$